The greedy algorithm for submodular function maximization subject to cardinality constraint is guaranteed to approximate the optimal solution to within a $1-1/e$ factor. For worst-case instances, it is well known that this guarantee is essentially tight -- for greedy and in fact any efficient algorithm. Motivated by the question of why greedy performs better in practice, we introduce a new notion of budget smoothed analysis. Our framework requires larger perturbations of budgets than traditional smoothed analysis for e.g. linear programming. Nevertheless, we show that under realistic budget distributions, greedy and related algorithms enjoy provably better approximation guarantees, that hold even for worst-case submodular functions.

翻译：对于最坏的情况,众所周知,这种保证基本上是严格的 -- -- 贪婪和实际上任何有效的算法。出于为什么贪婪在实践中表现得更好这一问题的动机,我们引入了新的预算平稳分析概念。我们的框架要求预算比传统的平滑分析对线性编程等项目进行更大的扰动。然而,我们表明,在现实的预算分配下,贪婪和相关算法享有甚至对最坏的子模式功能都维持的最好近似保证。